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Properties

Label 2-513-19.4-c1-0-16
Degree $2$
Conductor $513$
Sign $0.600 - 0.799i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.37 + 0.866i)2-s + (3.37 + 2.83i)4-s + (−1.03 + 0.866i)5-s + (2.43 − 4.22i)7-s + (3.05 + 5.28i)8-s + (−3.20 + 1.16i)10-s + (2.09 + 3.62i)11-s + (−0.460 + 2.61i)13-s + (9.46 − 7.94i)14-s + (1.15 + 6.53i)16-s + (−6.12 − 2.22i)17-s + (−2.52 − 3.55i)19-s − 5.94·20-s + (1.84 + 10.4i)22-s + (1.93 + 1.62i)23-s + ⋯
L(s)  = 1  + (1.68 + 0.612i)2-s + (1.68 + 1.41i)4-s + (−0.461 + 0.387i)5-s + (0.922 − 1.59i)7-s + (1.07 + 1.86i)8-s + (−1.01 + 0.368i)10-s + (0.630 + 1.09i)11-s + (−0.127 + 0.724i)13-s + (2.52 − 2.12i)14-s + (0.288 + 1.63i)16-s + (−1.48 − 0.540i)17-s + (−0.578 − 0.815i)19-s − 1.32·20-s + (0.392 + 2.22i)22-s + (0.404 + 0.339i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.600 - 0.799i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.23770 + 1.61783i\)
\(L(\frac12)\) \(\approx\) \(3.23770 + 1.61783i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (2.52 + 3.55i)T \)
good2 \( 1 + (-2.37 - 0.866i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (1.03 - 0.866i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-2.43 + 4.22i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.09 - 3.62i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.460 - 2.61i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (6.12 + 2.22i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-1.93 - 1.62i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.76 - 1.00i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.16 + 5.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.24T + 37T^{2} \)
41 \( 1 + (1.27 + 7.23i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (2.51 - 2.11i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-1.41 + 0.516i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-3.05 - 2.56i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-3.35 - 1.22i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.28 - 1.91i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-1.37 + 0.502i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (0.343 - 0.288i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.54 + 8.78i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.74 + 9.87i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.29 - 3.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.24 - 18.3i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-13.6 - 4.96i)T + (74.3 + 62.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.38274879242034583627551983712, −10.61178921801502356538712070802, −9.135264444281864073688785633056, −7.67919797668545111997151707782, −7.01708072736783942702801821480, −6.72089406401094780691543046490, −4.99922150333444514692251039772, −4.35577342568544802055126792905, −3.79356794785967610251351457027, −2.11341701229716773437361831072, 1.79158584017260240546184086360, 2.88984220361461505801704518098, 4.07678049239712886111938933476, 4.97424643121323846071363138452, 5.78565884828758528668498901264, 6.53046821789215927704955808827, 8.367844046990696849230362083971, 8.686903857823438099806217317881, 10.37255731092997192194542396488, 11.30798174830087290438854071316

Graph of the $Z$-function along the critical line